Computer Science > Data Structures and Algorithms

Title:A Sketching Algorithm for Spectral Graph Sparsification

Abstract: We study the problem of compressing a weighted graph $G$ on $n$ vertices,
building a "sketch" $H$ of $G$, so that given any vector $x \in \mathbb{R}^n$,
the value $x^T L_G x$ can be approximated up to a multiplicative $1+\epsilon$
factor from only $H$ and $x$, where $L_G$ denotes the Laplacian of $G$. One
solution to this problem is to build a spectral sparsifier $H$ of $G$, which,
using the result of Batson, Spielman, and Srivastava, consists of $O(n
\epsilon^{-2})$ reweighted edges of $G$ and has the property that
simultaneously for all $x \in \mathbb{R}^n$, $x^T L_H x = (1 \pm \epsilon) x^T
L_G x$. The $O(n \epsilon^{-2})$ bound is optimal for spectral sparsifiers. We
show that if one is interested in only preserving the value of $x^T L_G x$ for
a {\it fixed} $x \in \mathbb{R}^n$ (specified at query time) with high
probability, then there is a sketch $H$ using only $\tilde{O}(n
\epsilon^{-1.6})$ bits of space. This is the first data structure achieving a
sub-quadratic dependence on $\epsilon$. Our work builds upon recent work of
Andoni, Krauthgamer, and Woodruff who showed that $\tilde{O}(n \epsilon^{-1})$
bits of space is possible for preserving a fixed {\it cut query} (i.e., $x\in
\{0,1\}^n$) with high probability; here we show that even for a general query
vector $x \in \mathbb{R}^n$, a sub-quadratic dependence on $\epsilon$ is
possible. Our result for Laplacians is in sharp contrast to sketches for
general $n \times n$ positive semidefinite matrices $A$ with $O(\log n)$ bit
entries, for which even to preserve the value of $x^T A x$ for a fixed $x \in
\mathbb{R}^n$ (specified at query time) up to a $1+\epsilon$ factor with
constant probability, we show an $\Omega(n \epsilon^{-2})$ lower bound.